--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Hyperreal/Filter.ML Sat Dec 30 22:03:46 2000 +0100
@@ -0,0 +1,554 @@
+(* Title : Filter.ML
+ ID : $Id$
+ Author : Jacques D. Fleuriot
+ Copyright : 1998 University of Cambridge
+ Description : Filters and Ultrafilter
+*)
+
+(*------------------------------------------------------------------
+ Properties of Filters and Freefilters -
+ rules for intro, destruction etc.
+ ------------------------------------------------------------------*)
+
+Goalw [is_Filter_def] "is_Filter X S ==> X <= Pow(S)";
+by (Blast_tac 1);
+qed "is_FilterD1";
+
+Goalw [is_Filter_def] "is_Filter X S ==> X ~= {}";
+by (Blast_tac 1);
+qed "is_FilterD2";
+
+Goalw [is_Filter_def] "is_Filter X S ==> {} ~: X";
+by (Blast_tac 1);
+qed "is_FilterD3";
+
+Goalw [Filter_def] "is_Filter X S ==> X : Filter S";
+by (Blast_tac 1);
+qed "mem_FiltersetI";
+
+Goalw [Filter_def] "X : Filter S ==> is_Filter X S";
+by (Blast_tac 1);
+qed "mem_FiltersetD";
+
+Goal "X : Filter S ==> {} ~: X";
+by (etac (mem_FiltersetD RS is_FilterD3) 1);
+qed "Filter_empty_not_mem";
+
+bind_thm ("Filter_empty_not_memE",(Filter_empty_not_mem RS notE));
+
+Goalw [Filter_def,is_Filter_def]
+ "[| X: Filter S; A: X; B: X |] ==> A Int B : X";
+by (Blast_tac 1);
+qed "mem_FiltersetD1";
+
+Goalw [Filter_def,is_Filter_def]
+ "[| X: Filter S; A: X; A <= B; B <= S|] ==> B : X";
+by (Blast_tac 1);
+qed "mem_FiltersetD2";
+
+Goalw [Filter_def,is_Filter_def]
+ "[| X: Filter S; A: X |] ==> A : Pow S";
+by (Blast_tac 1);
+qed "mem_FiltersetD3";
+
+Goalw [Filter_def,is_Filter_def]
+ "X: Filter S ==> S : X";
+by (Blast_tac 1);
+qed "mem_FiltersetD4";
+
+Goalw [is_Filter_def]
+ "[| X <= Pow(S);\
+\ S : X; \
+\ X ~= {}; \
+\ {} ~: X; \
+\ ALL u: X. ALL v: X. u Int v : X; \
+\ ALL u v. u: X & u<=v & v<=S --> v: X \
+\ |] ==> is_Filter X S";
+by (Blast_tac 1);
+qed "is_FilterI";
+
+Goal "[| X <= Pow(S);\
+\ S : X; \
+\ X ~= {}; \
+\ {} ~: X; \
+\ ALL u: X. ALL v: X. u Int v : X; \
+\ ALL u v. u: X & u<=v & v<=S --> v: X \
+\ |] ==> X: Filter S";
+by (blast_tac (claset() addIs [mem_FiltersetI,is_FilterI]) 1);
+qed "mem_FiltersetI2";
+
+Goalw [is_Filter_def]
+ "is_Filter X S ==> X <= Pow(S) & \
+\ S : X & \
+\ X ~= {} & \
+\ {} ~: X & \
+\ (ALL u: X. ALL v: X. u Int v : X) & \
+\ (ALL u v. u: X & u <= v & v<=S --> v: X)";
+by (Fast_tac 1);
+qed "is_FilterE_lemma";
+
+Goalw [is_Filter_def]
+ "X : Filter S ==> X <= Pow(S) &\
+\ S : X & \
+\ X ~= {} & \
+\ {} ~: X & \
+\ (ALL u: X. ALL v: X. u Int v : X) & \
+\ (ALL u v. u: X & u <= v & v<=S --> v: X)";
+by (etac (mem_FiltersetD RS is_FilterE_lemma) 1);
+qed "memFiltersetE_lemma";
+
+Goalw [Filter_def,Freefilter_def]
+ "X: Freefilter S ==> X: Filter S";
+by (Fast_tac 1);
+qed "Freefilter_Filter";
+
+Goalw [Freefilter_def]
+ "X: Freefilter S ==> ALL y: X. ~finite(y)";
+by (Blast_tac 1);
+qed "mem_Freefilter_not_finite";
+
+Goal "[| X: Freefilter S; x: X |] ==> ~ finite x";
+by (blast_tac (claset() addSDs [mem_Freefilter_not_finite]) 1);
+qed "mem_FreefiltersetD1";
+
+bind_thm ("mem_FreefiltersetE1", (mem_FreefiltersetD1 RS notE));
+
+Goal "[| X: Freefilter S; finite x|] ==> x ~: X";
+by (blast_tac (claset() addSDs [mem_Freefilter_not_finite]) 1);
+qed "mem_FreefiltersetD2";
+
+Goalw [Freefilter_def]
+ "[| X: Filter S; ALL x. ~(x: X & finite x) |] ==> X: Freefilter S";
+by (Blast_tac 1);
+qed "mem_FreefiltersetI1";
+
+Goalw [Freefilter_def]
+ "[| X: Filter S; ALL x. (x ~: X | ~ finite x) |] ==> X: Freefilter S";
+by (Blast_tac 1);
+qed "mem_FreefiltersetI2";
+
+Goal "[| X: Filter S; A: X; B: X |] ==> A Int B ~= {}";
+by (forw_inst_tac [("A","A"),("B","B")] mem_FiltersetD1 1);
+by (auto_tac (claset() addSDs [Filter_empty_not_mem],simpset()));
+qed "Filter_Int_not_empty";
+
+bind_thm ("Filter_Int_not_emptyE",(Filter_Int_not_empty RS notE));
+
+(*----------------------------------------------------------------------------------
+ Ultrafilters and Free ultrafilters
+ ----------------------------------------------------------------------------------*)
+
+Goalw [Ultrafilter_def] "X : Ultrafilter S ==> X: Filter S";
+by (Blast_tac 1);
+qed "Ultrafilter_Filter";
+
+Goalw [Ultrafilter_def]
+ "X : Ultrafilter S ==> !A: Pow(S). A : X | S - A: X";
+by (Blast_tac 1);
+qed "mem_UltrafiltersetD2";
+
+Goalw [Ultrafilter_def]
+ "[|X : Ultrafilter S; A <= S; A ~: X |] ==> S - A: X";
+by (Blast_tac 1);
+qed "mem_UltrafiltersetD3";
+
+Goalw [Ultrafilter_def]
+ "[|X : Ultrafilter S; A <= S; S - A ~: X |] ==> A: X";
+by (Blast_tac 1);
+qed "mem_UltrafiltersetD4";
+
+Goalw [Ultrafilter_def]
+ "[| X: Filter S; \
+\ ALL A: Pow(S). A: X | S - A : X |] ==> X: Ultrafilter S";
+by (Blast_tac 1);
+qed "mem_UltrafiltersetI";
+
+Goalw [Ultrafilter_def,FreeUltrafilter_def]
+ "X: FreeUltrafilter S ==> X: Ultrafilter S";
+by (Blast_tac 1);
+qed "FreeUltrafilter_Ultrafilter";
+
+Goalw [FreeUltrafilter_def]
+ "X: FreeUltrafilter S ==> ALL y: X. ~finite(y)";
+by (Blast_tac 1);
+qed "mem_FreeUltrafilter_not_finite";
+
+Goal "[| X: FreeUltrafilter S; x: X |] ==> ~ finite x";
+by (blast_tac (claset() addSDs [mem_FreeUltrafilter_not_finite]) 1);
+qed "mem_FreeUltrafiltersetD1";
+
+bind_thm ("mem_FreeUltrafiltersetE1", (mem_FreeUltrafiltersetD1 RS notE));
+
+Goal "[| X: FreeUltrafilter S; finite x|] ==> x ~: X";
+by (blast_tac (claset() addSDs [mem_FreeUltrafilter_not_finite]) 1);
+qed "mem_FreeUltrafiltersetD2";
+
+Goalw [FreeUltrafilter_def]
+ "[| X: Ultrafilter S; \
+\ ALL x. ~(x: X & finite x) |] ==> X: FreeUltrafilter S";
+by (Blast_tac 1);
+qed "mem_FreeUltrafiltersetI1";
+
+Goalw [FreeUltrafilter_def]
+ "[| X: Ultrafilter S; \
+\ ALL x. (x ~: X | ~ finite x) |] ==> X: FreeUltrafilter S";
+by (Blast_tac 1);
+qed "mem_FreeUltrafiltersetI2";
+
+Goalw [FreeUltrafilter_def,Freefilter_def,Ultrafilter_def]
+ "(X: FreeUltrafilter S) = (X: Freefilter S & (ALL x:Pow(S). x: X | S - x: X))";
+by (Blast_tac 1);
+qed "FreeUltrafilter_iff";
+
+(*-------------------------------------------------------------------
+ A Filter F on S is an ultrafilter iff it is a maximal filter
+ i.e. whenever G is a filter on I and F <= F then F = G
+ --------------------------------------------------------------------*)
+(*---------------------------------------------------------------------
+ lemmas that shows existence of an extension to what was assumed to
+ be a maximal filter. Will be used to derive contradiction in proof of
+ property of ultrafilter
+ ---------------------------------------------------------------------*)
+Goal "[| F ~= {}; A <= S |] ==> \
+\ EX x. x: {X. X <= S & (EX f:F. A Int f <= X)}";
+by (Blast_tac 1);
+qed "lemma_set_extend";
+
+Goal "a: X ==> X ~= {}";
+by (Step_tac 1);
+qed "lemma_set_not_empty";
+
+Goal "x Int F <= {} ==> F <= - x";
+by (Blast_tac 1);
+qed "lemma_empty_Int_subset_Compl";
+
+Goalw [Filter_def,is_Filter_def]
+ "[| F: Filter S; A ~: F; A <= S|] \
+\ ==> ALL B. B ~: F | ~ B <= A";
+by (Blast_tac 1);
+qed "mem_Filterset_disjI";
+
+Goal "F : Ultrafilter S ==> \
+\ (F: Filter S & (ALL G: Filter S. F <= G --> F = G))";
+by (auto_tac (claset(),simpset() addsimps [Ultrafilter_def]));
+by (dres_inst_tac [("x","x")] bspec 1);
+by (etac mem_FiltersetD3 1 THEN assume_tac 1);
+by (Step_tac 1);
+by (dtac subsetD 1 THEN assume_tac 1);
+by (blast_tac (claset() addSDs [Filter_Int_not_empty]) 1);
+qed "Ultrafilter_max_Filter";
+
+
+(*--------------------------------------------------------------------------------
+ This is a very long and tedious proof; need to break it into parts.
+ Have proof that {X. X <= S & (EX f: F. A Int f <= X)} is a filter as
+ a lemma
+--------------------------------------------------------------------------------*)
+Goalw [Ultrafilter_def]
+ "[| F: Filter S; \
+\ ALL G: Filter S. F <= G --> F = G |] ==> F : Ultrafilter S";
+by (Step_tac 1);
+by (rtac ccontr 1);
+by (forward_tac [mem_FiltersetD RS is_FilterD2] 1);
+by (forw_inst_tac [("x","{X. X <= S & (EX f: F. A Int f <= X)}")] bspec 1);
+by (EVERY1[rtac mem_FiltersetI2, Blast_tac, Asm_full_simp_tac]);
+by (blast_tac (claset() addDs [mem_FiltersetD3]) 1);
+by (etac (lemma_set_extend RS exE) 1);
+by (assume_tac 1 THEN etac lemma_set_not_empty 1);
+by (REPEAT(rtac ballI 2) THEN Asm_full_simp_tac 2);
+by (rtac conjI 2 THEN Blast_tac 2);
+by (REPEAT(etac conjE 2) THEN REPEAT(etac bexE 2));
+by (res_inst_tac [("x","f Int fa")] bexI 2);
+by (etac mem_FiltersetD1 3);
+by (assume_tac 3 THEN assume_tac 3);
+by (Fast_tac 2);
+by (EVERY[REPEAT(rtac allI 2), rtac impI 2,Asm_full_simp_tac 2]);
+by (EVERY[REPEAT(etac conjE 2), etac bexE 2]);
+by (res_inst_tac [("x","f")] bexI 2);
+by (rtac subsetI 2);
+by (Fast_tac 2 THEN assume_tac 2);
+by (Step_tac 2);
+by (Blast_tac 3);
+by (eres_inst_tac [("c","A")] equalityCE 3);
+by (REPEAT(Blast_tac 3));
+by (dres_inst_tac [("A","xa")] mem_FiltersetD3 2 THEN assume_tac 2);
+by (Blast_tac 2);
+by (dtac lemma_empty_Int_subset_Compl 1);
+by (EVERY1[ftac mem_Filterset_disjI , assume_tac, Fast_tac]);
+by (dtac mem_FiltersetD3 1 THEN assume_tac 1);
+by (dres_inst_tac [("x","f")] spec 1);
+by (Blast_tac 1);
+qed "max_Filter_Ultrafilter";
+
+Goal "(F : Ultrafilter S) = (F: Filter S & (ALL G: Filter S. F <= G --> F = G))";
+by (blast_tac (claset() addSIs [Ultrafilter_max_Filter,max_Filter_Ultrafilter]) 1);
+qed "Ultrafilter_iff";
+
+(*--------------------------------------------------------------------
+ A few properties of freefilters
+ -------------------------------------------------------------------*)
+
+Goal "F1 Int F2 = ((F1 Int Y) Int F2) Un ((F2 Int (- Y)) Int F1)";
+by (Auto_tac);
+qed "lemma_Compl_cancel_eq";
+
+Goal "finite X ==> finite (X Int Y)";
+by (etac (Int_lower1 RS finite_subset) 1);
+qed "finite_IntI1";
+
+Goal "finite Y ==> finite (X Int Y)";
+by (etac (Int_lower2 RS finite_subset) 1);
+qed "finite_IntI2";
+
+Goal "[| finite (F1 Int Y); \
+\ finite (F2 Int (- Y)) \
+\ |] ==> finite (F1 Int F2)";
+by (res_inst_tac [("Y1","Y")] (lemma_Compl_cancel_eq RS ssubst) 1);
+by (rtac finite_UnI 1);
+by (auto_tac (claset() addSIs [finite_IntI1,finite_IntI2],simpset()));
+qed "finite_Int_Compl_cancel";
+
+Goal "U: Freefilter S ==> \
+\ ~ (EX f1: U. EX f2: U. finite (f1 Int x) \
+\ & finite (f2 Int (- x)))";
+by (Step_tac 1);
+by (forw_inst_tac [("A","f1"),("B","f2")]
+ (Freefilter_Filter RS mem_FiltersetD1) 1);
+by (dres_inst_tac [("x","f1 Int f2")] mem_FreefiltersetD1 3);
+by (dtac finite_Int_Compl_cancel 4);
+by (Auto_tac);
+qed "Freefilter_lemma_not_finite";
+
+(* the lemmas below follow *)
+Goal "U: Freefilter S ==> \
+\ ALL f: U. ~ finite (f Int x) | ~finite (f Int (- x))";
+by (blast_tac (claset() addSDs [Freefilter_lemma_not_finite,bspec]) 1);
+qed "Freefilter_Compl_not_finite_disjI";
+
+Goal "U: Freefilter S ==> \
+\ (ALL f: U. ~ finite (f Int x)) | (ALL f:U. ~finite (f Int (- x)))";
+by (blast_tac (claset() addSDs [Freefilter_lemma_not_finite,bspec]) 1);
+qed "Freefilter_Compl_not_finite_disjI2";
+
+Goal "- UNIV = {}";
+by (Auto_tac );
+qed "Compl_UNIV_eq";
+
+Addsimps [Compl_UNIV_eq];
+
+Goal "- {} = UNIV";
+by (Auto_tac );
+qed "Compl_empty_eq";
+
+Addsimps [Compl_empty_eq];
+
+val [prem] = goal (the_context ()) "~ finite (UNIV:: 'a set) ==> \
+\ {A:: 'a set. finite (- A)} : Filter UNIV";
+by (cut_facts_tac [prem] 1);
+by (rtac mem_FiltersetI2 1);
+by (auto_tac (claset(), simpset() delsimps [Collect_empty_eq]));
+by (eres_inst_tac [("c","UNIV")] equalityCE 1);
+by (Auto_tac);
+by (etac (Compl_anti_mono RS finite_subset) 1);
+by (assume_tac 1);
+qed "cofinite_Filter";
+
+Goal "~finite(UNIV :: 'a set) ==> ~finite (X :: 'a set) | ~finite (- X)";
+by (dres_inst_tac [("A1","X")] (Compl_partition RS ssubst) 1);
+by (Asm_full_simp_tac 1);
+qed "not_finite_UNIV_disjI";
+
+Goal "[| ~finite(UNIV :: 'a set); \
+\ finite (X :: 'a set) \
+\ |] ==> ~finite (- X)";
+by (dres_inst_tac [("X","X")] not_finite_UNIV_disjI 1);
+by (Blast_tac 1);
+qed "not_finite_UNIV_Compl";
+
+val [prem] = goal (the_context ()) "~ finite (UNIV:: 'a set) ==> \
+\ !X: {A:: 'a set. finite (- A)}. ~ finite X";
+by (cut_facts_tac [prem] 1);
+by (auto_tac (claset() addDs [not_finite_UNIV_disjI],simpset()));
+qed "mem_cofinite_Filter_not_finite";
+
+val [prem] = goal (the_context ()) "~ finite (UNIV:: 'a set) ==> \
+\ {A:: 'a set. finite (- A)} : Freefilter UNIV";
+by (cut_facts_tac [prem] 1);
+by (rtac mem_FreefiltersetI2 1);
+by (rtac cofinite_Filter 1 THEN assume_tac 1);
+by (blast_tac (claset() addSDs [mem_cofinite_Filter_not_finite]) 1);
+qed "cofinite_Freefilter";
+
+Goal "UNIV - x = - x";
+by (Auto_tac);
+qed "UNIV_diff_Compl";
+Addsimps [UNIV_diff_Compl];
+
+Goalw [Ultrafilter_def,FreeUltrafilter_def]
+ "[| ~finite(UNIV :: 'a set); (U :: 'a set set): FreeUltrafilter UNIV\
+\ |] ==> {X. finite(- X)} <= U";
+by (ftac cofinite_Filter 1);
+by (Step_tac 1);
+by (forw_inst_tac [("X","- x :: 'a set")] not_finite_UNIV_Compl 1);
+by (assume_tac 1);
+by (Step_tac 1 THEN Fast_tac 1);
+by (dres_inst_tac [("x","x")] bspec 1);
+by (Blast_tac 1);
+by (asm_full_simp_tac (simpset() addsimps [UNIV_diff_Compl]) 1);
+qed "FreeUltrafilter_contains_cofinite_set";
+
+(*--------------------------------------------------------------------
+ We prove: 1. Existence of maximal filter i.e. ultrafilter
+ 2. Freeness property i.e ultrafilter is free
+ Use a locale to prove various lemmas and then
+ export main result: The Ultrafilter Theorem
+ -------------------------------------------------------------------*)
+Open_locale "UFT";
+
+Goalw [chain_def, thm "superfrechet_def", thm "frechet_def"]
+ "!!(c :: 'a set set set). c : chain (superfrechet S) ==> Union c <= Pow S";
+by (Step_tac 1);
+by (dtac subsetD 1 THEN assume_tac 1);
+by (Step_tac 1);
+by (dres_inst_tac [("X","X")] mem_FiltersetD3 1);
+by (Auto_tac);
+qed "chain_Un_subset_Pow";
+
+Goalw [chain_def,Filter_def,is_Filter_def,
+ thm "superfrechet_def", thm "frechet_def"]
+ "!!(c :: 'a set set set). c: chain (superfrechet S) \
+\ ==> !x: c. {} < x";
+by (blast_tac (claset() addSIs [psubsetI]) 1);
+qed "mem_chain_psubset_empty";
+
+Goal "!!(c :: 'a set set set). \
+\ [| c: chain (superfrechet S);\
+\ c ~= {} \
+\ |]\
+\ ==> Union(c) ~= {}";
+by (dtac mem_chain_psubset_empty 1);
+by (Step_tac 1);
+by (dtac bspec 1 THEN assume_tac 1);
+by (auto_tac (claset() addDs [Union_upper,bspec],
+ simpset() addsimps [psubset_def]));
+qed "chain_Un_not_empty";
+
+Goalw [is_Filter_def,Filter_def,chain_def,thm "superfrechet_def"]
+ "!!(c :: 'a set set set). \
+\ c : chain (superfrechet S) \
+\ ==> {} ~: Union(c)";
+by (Blast_tac 1);
+qed "Filter_empty_not_mem_Un";
+
+Goal "c: chain (superfrechet S) \
+\ ==> ALL u : Union(c). ALL v: Union(c). u Int v : Union(c)";
+by (Step_tac 1);
+by (forw_inst_tac [("x","X"),("y","Xa")] chainD 1);
+by (REPEAT(assume_tac 1));
+by (dtac chainD2 1);
+by (etac disjE 1);
+by (res_inst_tac [("X","Xa")] UnionI 1 THEN assume_tac 1);
+by (dres_inst_tac [("A","X")] subsetD 1 THEN assume_tac 1);
+by (dres_inst_tac [("c","Xa")] subsetD 1 THEN assume_tac 1);
+by (res_inst_tac [("X","X")] UnionI 2 THEN assume_tac 2);
+by (dres_inst_tac [("A","Xa")] subsetD 2 THEN assume_tac 2);
+by (dres_inst_tac [("c","X")] subsetD 2 THEN assume_tac 2);
+by (auto_tac (claset() addIs [mem_FiltersetD1],
+ simpset() addsimps [thm "superfrechet_def"]));
+qed "Filter_Un_Int";
+
+Goal "c: chain (superfrechet S) \
+\ ==> ALL u v. u: Union(c) & \
+\ (u :: 'a set) <= v & v <= S --> v: Union(c)";
+by (Step_tac 1);
+by (dtac chainD2 1);
+by (dtac subsetD 1 THEN assume_tac 1);
+by (rtac UnionI 1 THEN assume_tac 1);
+by (auto_tac (claset() addIs [mem_FiltersetD2],
+ simpset() addsimps [thm "superfrechet_def"]));
+qed "Filter_Un_subset";
+
+Goalw [chain_def,thm "superfrechet_def"]
+ "!!(c :: 'a set set set). \
+\ [| c: chain (superfrechet S);\
+\ x: c \
+\ |] ==> x : Filter S";
+by (Blast_tac 1);
+qed "lemma_mem_chain_Filter";
+
+Goalw [chain_def,thm "superfrechet_def"]
+ "!!(c :: 'a set set set). \
+\ [| c: chain (superfrechet S);\
+\ x: c \
+\ |] ==> frechet S <= x";
+by (Blast_tac 1);
+qed "lemma_mem_chain_frechet_subset";
+
+Goal "!!(c :: 'a set set set). \
+\ [| c ~= {}; \
+\ c : chain (superfrechet (UNIV :: 'a set))\
+\ |] ==> Union c : superfrechet (UNIV)";
+by (simp_tac (simpset() addsimps
+ [thm "superfrechet_def",thm "frechet_def"]) 1);
+by (Step_tac 1);
+by (rtac mem_FiltersetI2 1);
+by (etac chain_Un_subset_Pow 1);
+by (rtac UnionI 1 THEN assume_tac 1);
+by (etac (lemma_mem_chain_Filter RS mem_FiltersetD4) 1 THEN assume_tac 1);
+by (etac chain_Un_not_empty 1);
+by (etac Filter_empty_not_mem_Un 2);
+by (etac Filter_Un_Int 2);
+by (etac Filter_Un_subset 2);
+by (subgoal_tac "xa : frechet (UNIV)" 2);
+by (rtac UnionI 2 THEN assume_tac 2);
+by (rtac (lemma_mem_chain_frechet_subset RS subsetD) 2);
+by (auto_tac (claset(),simpset() addsimps [thm "frechet_def"]));
+qed "Un_chain_mem_cofinite_Filter_set";
+
+Goal "EX U: superfrechet (UNIV). \
+\ ALL G: superfrechet (UNIV). U <= G --> U = G";
+by (rtac Zorn_Lemma2 1);
+by (cut_facts_tac [thm "not_finite_UNIV" RS cofinite_Filter] 1);
+by (Step_tac 1);
+by (res_inst_tac [("Q","c={}")] (excluded_middle RS disjE) 1);
+by (res_inst_tac [("x","Union c")] bexI 1 THEN Blast_tac 1);
+by (rtac Un_chain_mem_cofinite_Filter_set 1 THEN REPEAT(assume_tac 1));
+by (res_inst_tac [("x","frechet (UNIV)")] bexI 1 THEN Blast_tac 1);
+by (auto_tac (claset(),
+ simpset() addsimps
+ [thm "superfrechet_def", thm "frechet_def"]));
+qed "max_cofinite_Filter_Ex";
+
+Goal "EX U: superfrechet UNIV. (\
+\ ALL G: superfrechet UNIV. U <= G --> U = G) \
+\ & (ALL x: U. ~finite x)";
+by (cut_facts_tac [thm "not_finite_UNIV" RS
+ (export max_cofinite_Filter_Ex)] 1);
+by (Step_tac 1);
+by (res_inst_tac [("x","U")] bexI 1);
+by (auto_tac (claset(),simpset() addsimps
+ [thm "superfrechet_def", thm "frechet_def"]));
+by (dres_inst_tac [("c","- x")] subsetD 1);
+by (Asm_simp_tac 1);
+by (forw_inst_tac [("A","x"),("B","- x")] mem_FiltersetD1 1);
+by (dtac Filter_empty_not_mem 3);
+by (ALLGOALS(Asm_full_simp_tac ));
+qed "max_cofinite_Freefilter_Ex";
+
+(*--------------------------------------------------------------------------------
+ There exists a free ultrafilter on any infinite set
+ --------------------------------------------------------------------------------*)
+
+Goalw [FreeUltrafilter_def] "EX U. U: FreeUltrafilter (UNIV :: 'a set)";
+by (cut_facts_tac [thm "not_finite_UNIV" RS (export max_cofinite_Freefilter_Ex)] 1);
+by (asm_full_simp_tac (simpset() addsimps
+ [thm "superfrechet_def", Ultrafilter_iff, thm "frechet_def"]) 1);
+by (Step_tac 1);
+by (res_inst_tac [("x","U")] exI 1);
+by (Step_tac 1);
+by (Blast_tac 1);
+qed "FreeUltrafilter_ex";
+
+bind_thm ("FreeUltrafilter_Ex", export FreeUltrafilter_ex);
+
+Close_locale "UFT";